Pushdown and Lempel-Ziv Depth

Abstract

This paper expands upon existing and introduces new formulations of Bennett's logical depth. In previously published work by Jordon and Moser, notions of finite-state-depth and pushdown-depth were examined and compared. These were based on finite-state transducers and information lossless pushdown compressors respectively. Unfortunately a full separation between the two notions was not established. This paper introduces a new formulation of pushdown-depth based on restricting how fast a pushdown compressor's stack can grow. This improved formulation allows us to do a full comparison by demonstrating the existence of sequences with high finite-state-depth and low pushdown-depth, and vice-versa. A new notion based on the Lempel-Ziv '78 algorithm is also introduced. Its difference from finite-state-depth is shown by demonstrating the existence of a Lempel-Ziv deep sequence that is not finitestate deep and vice versa. Lempel-Ziv-depth's difference from pushdown-depth is shown by building sequences that have a pushdown-depth of roughly 1/2 but low Lempel-Ziv depth, and a sequence with high Lempel-Ziv depth but low pushdown-depth. Properties of all three notions are also discussed and proved.

Pebble-Depth

“…It is chosen as it is required in the proof of lemma 2.6 which requires an upperbound for the size of FSTs with the same transition and output functions, but with different start states. We omit the full details of the representation as if a sequence is FS-deep with respect to one representation, it is FS-deep with respect to all representations [17]. Hence we write D k FS (x) to represent the k-FS complexity of string x.…”

“…We compare PB-depth with i.o. finite-state-depth (FS-depth) which is based on finite-state-transducers [9], pushdown-depth (PD-depth) which is based on lossless pushdown compressors whose stack is bounded by height O(log n) on inputs of length n [17], and Lempel-Ziv-depth (LZ-depth) [17] based on the lossless compression algorithm Lempel-Ziv 78 [29]. To compare PB-depth with FS-depth and PD-depth, we build a sequence S that has a PB-depth level of approximitately 1, PD-depth level of approximately 1/2 and FS-depth close to 0.…”